That is valid, because the Minkowski metric is constant (so [itex]\partial_\epsilon \eta^{\sigma \epsilon} = 0[/itex]). However, for general metrics g, [itex]\partial_\alpha g^{\alpha \beta}[/itex] is not necessarily zero. The problem disappears if you replace partial derivatives with covariant derivatives, though, because (by definition of the Levi-Civita connection) in this case [itex]\nabla_\alpha g^{\alpha \beta} = 0[/itex].